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          线性代数
        </a></li> <!----></ul> <div class="info" data-v-18fb2c02><div title="作者" class="author iconfont icon-touxiang" data-v-18fb2c02><a href="javascript:;" data-v-18fb2c02>aiyoudiao</a></div> <div title="创建时间" class="date iconfont icon-riqi" data-v-18fb2c02><a href="javascript:;" data-v-18fb2c02>2022-05-02</a></div> <!----></div></div></div> <!----> <div class="content-wrapper"><div class="right-menu-wrapper"><div class="right-menu-margin"><div class="right-menu-content"></div></div></div> <h1><!----> <span>
            高斯消元基础
          </span></h1> <div class="theme-vdoing-content content__default"><h2 id="消元法"><a href="#消元法" class="header-anchor">#</a> 消元法</h2> <p>消元法来解线性方程组，通过不停的消减未知数，最终得到所有未知数的值。</p> <p>常用的方式有三种基本操作：</p> <ol><li>一个方程的左右两边同时乘以一个常数。</li> <li>用一个方程加或减另一个方程组。</li> <li>交换两个方程的位置</li></ol> <h2 id="增广矩阵"><a href="#增广矩阵" class="header-anchor">#</a> 增广矩阵</h2> <p>行数代表方程的个数，列代表方程的未知数。</p> <p>在原来系数矩阵的基础上添加了一列结果列，那么这个矩阵就叫做增广矩阵。</p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651476165461-5e2bc2bd-adb0-45c1-8baa-7b6912918fd5.png" alt=""></p> <h2 id="运算过程"><a href="#运算过程" class="header-anchor">#</a> 运算过程</h2> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651476344851-cc328275-3853-4fe9-b59f-e5d6e1cb074d.png" alt=""></p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651476431559-ee4a2c4a-642d-46a2-b4fa-9fb18da26d04.png" alt=""></p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651479563515-40e0871b-9d40-4dd3-886f-f826f1ea5a54.png" alt=""></p> <h2 id="消元操作转矩阵操作"><a href="#消元操作转矩阵操作" class="header-anchor">#</a> 消元操作转矩阵操作</h2> <p><strong>一个方程的左右两边同时乘以一个常数</strong></p> <p>矩阵中某一行乘以一个常数</p> <p><strong>用一个方程加或减另一个方程组</strong></p> <p>矩阵中的一行加或减另一行</p> <p><strong>交换两个方程的位置</strong></p> <p>交换矩阵的两行</p> <h2 id="高斯消元"><a href="#高斯消元" class="header-anchor">#</a> 高斯消元</h2> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651479898726-e63e2584-74e3-43d0-a4eb-a6468920d26e.png" alt=""></p> <p>主元就是每一行主要的系数。</p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651479986855-8b03ea16-a59b-4395-a58b-f8138cbc8d27.png" alt=""></p> <p>将第n行的第n个元素的系数变为1。</p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651480182123-d8cf9e50-5288-4437-8da1-a6d7265febed.png" alt=""></p> <p>有时会涉及到两行交换。</p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651480514646-7018a798-22d2-4d6f-ae1b-5f7b6588b56e.png" alt=""></p> <p>甚至还有向前回代，但也可以无回代，那就是高斯约旦消元法</p> <h2 id="高斯约旦消元法"><a href="#高斯约旦消元法" class="header-anchor">#</a> 高斯约旦消元法</h2> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651480703649-635e6bbe-afa5-43f4-a86a-e471694d5af9.png" alt=""></p> <p>让第二行 加上 第三行乘以10，就能求出y了。</p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651480865881-de5b28d7-0372-41a8-a5eb-9dbc5d850ba4.png" alt=""></p> <p>让第 减去 四乘以第三行</p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651480907342-2da21c17-3586-4c07-984c-808eee848e25.png" alt=""> <img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651480943475-17b2f206-9b59-4562-b7db-1571ebed118b.png" alt=""></p> <p>让第一行 减去 二倍的第二行</p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651481010460-ecf4189b-b4e8-41fa-9005-bfd09a968aae.png" alt=""></p> <p>最终结果，这就是逆序的再进行一下消元。
<img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651481026487-ffabeb59-6a46-46d0-acae-a2c80ee89f80.png" alt=""></p> <p>除了主元以外的其它元都被削去，这就是高斯约旦消元法。</p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651481098849-f8109d6f-6468-467a-9c2c-afbb900bdedf.png" alt=""></p> <p>高斯消元法只有一个前向的过程，也就是从上到下。而高斯约旦消元法完成了后向过程，也就是从下往上。</p> <p><img src="https://gitee.com/aiyoudiao/images/raw/master/2022/05/02/1651481267846-c6c7bdc3-5c2d-4470-8555-b89dd0a577aa.png" alt=""></p> <h2 id="代码"><a href="#代码" class="header-anchor">#</a> 代码</h2> <div class="language-js extra-class"><pre class="language-js"><code><span class="token keyword">class</span> <span class="token class-name">LinearSystem</span> <span class="token punctuation">{</span>
    <span class="token function">constructor</span><span class="token punctuation">(</span><span class="token parameter"><span class="token constant">A</span><span class="token punctuation">,</span> b</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
        console<span class="token punctuation">.</span><span class="token function">assert</span><span class="token punctuation">(</span><span class="token constant">A</span><span class="token punctuation">.</span>length <span class="token operator">===</span> b<span class="token punctuation">.</span>length<span class="token punctuation">,</span> <span class="token string">&quot;矩阵行数和值的个数得一样&quot;</span><span class="token punctuation">)</span>
        <span class="token keyword">this</span><span class="token punctuation">.</span>m <span class="token operator">=</span> <span class="token constant">A</span><span class="token punctuation">.</span>length
        <span class="token keyword">this</span><span class="token punctuation">.</span>n <span class="token operator">=</span> <span class="token constant">A</span><span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length

        console<span class="token punctuation">.</span><span class="token function">assert</span><span class="token punctuation">(</span><span class="token keyword">this</span><span class="token punctuation">.</span>m<span class="token operator">===</span><span class="token keyword">this</span><span class="token punctuation">.</span>n<span class="token punctuation">,</span> <span class="token string">'矩阵的行列必须相同'</span><span class="token punctuation">)</span>

        <span class="token keyword">this</span><span class="token punctuation">.</span>Ab <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token punctuation">]</span> <span class="token comment">// 增广矩阵</span>
        <span class="token constant">A</span><span class="token punctuation">.</span><span class="token function">forEach</span><span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token parameter">row<span class="token punctuation">,</span> i</span><span class="token punctuation">)</span> <span class="token operator">=&gt;</span> <span class="token punctuation">{</span>
            <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">.</span><span class="token function">push</span><span class="token punctuation">(</span><span class="token punctuation">[</span><span class="token operator">...</span>row<span class="token punctuation">,</span> b<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">]</span><span class="token punctuation">)</span>
        <span class="token punctuation">}</span><span class="token punctuation">)</span>

        <span class="token comment">// 备份一下原方程</span>
        <span class="token keyword">this</span><span class="token punctuation">.</span>originAb <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token punctuation">]</span>
        <span class="token constant">A</span><span class="token punctuation">.</span><span class="token function">forEach</span><span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token parameter">row<span class="token punctuation">,</span> i</span><span class="token punctuation">)</span> <span class="token operator">=&gt;</span> <span class="token punctuation">{</span>
            <span class="token keyword">this</span><span class="token punctuation">.</span>originAb<span class="token punctuation">.</span><span class="token function">push</span><span class="token punctuation">(</span><span class="token punctuation">[</span><span class="token operator">...</span>row<span class="token punctuation">,</span> b<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">]</span><span class="token punctuation">)</span>
        <span class="token punctuation">}</span><span class="token punctuation">)</span>
    <span class="token punctuation">}</span>

    <span class="token function">gaussJordanElimination</span><span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>

        <span class="token keyword">this</span><span class="token punctuation">.</span><span class="token function">_forward</span><span class="token punctuation">(</span><span class="token punctuation">)</span>

        <span class="token keyword">this</span><span class="token punctuation">.</span><span class="token function">_backward</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
    <span class="token punctuation">}</span>

    <span class="token function">fancyPrint</span> <span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>

        <span class="token keyword">const</span> origin <span class="token operator">=</span> <span class="token keyword">this</span><span class="token punctuation">.</span><span class="token function">_fancyPrint</span><span class="token punctuation">(</span><span class="token keyword">this</span><span class="token punctuation">.</span>originAb<span class="token punctuation">)</span>
        <span class="token keyword">const</span> result <span class="token operator">=</span> <span class="token keyword">this</span><span class="token punctuation">.</span><span class="token function">_fancyPrint</span><span class="token punctuation">(</span><span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">)</span>
        console<span class="token punctuation">.</span><span class="token function">log</span><span class="token punctuation">(</span><span class="token string">&quot;============= 原方程 =============&quot;</span><span class="token punctuation">)</span>
        console<span class="token punctuation">.</span><span class="token function">log</span><span class="token punctuation">(</span>origin<span class="token punctuation">)</span>
        console<span class="token punctuation">.</span><span class="token function">log</span><span class="token punctuation">(</span><span class="token string">&quot;=========消元后方程结果：=========&quot;</span><span class="token punctuation">)</span>
        console<span class="token punctuation">.</span><span class="token function">log</span><span class="token punctuation">(</span>result<span class="token punctuation">)</span>
    <span class="token punctuation">}</span>

    <span class="token function">_fancyPrint</span> <span class="token punctuation">(</span><span class="token parameter">Ab</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">const</span> result <span class="token operator">=</span> <span class="token function">Array</span><span class="token punctuation">(</span><span class="token keyword">this</span><span class="token punctuation">.</span>m<span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">fill</span><span class="token punctuation">(</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">map</span><span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token parameter">_$i<span class="token punctuation">,</span>i</span><span class="token punctuation">)</span> <span class="token operator">=&gt;</span> <span class="token punctuation">{</span>
            <span class="token keyword">let</span> content <span class="token operator">=</span> <span class="token template-string"><span class="token template-punctuation string">`</span><span class="token template-punctuation string">`</span></span>
            content <span class="token operator">=</span> <span class="token function">Array</span><span class="token punctuation">(</span><span class="token keyword">this</span><span class="token punctuation">.</span>n<span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">fill</span><span class="token punctuation">(</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">map</span><span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token parameter">_$j<span class="token punctuation">,</span> j</span><span class="token punctuation">)</span> <span class="token operator">=&gt;</span> <span class="token punctuation">{</span>
                <span class="token keyword">return</span> Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>j<span class="token punctuation">]</span>
            <span class="token punctuation">}</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">join</span><span class="token punctuation">(</span><span class="token string">' '</span><span class="token punctuation">)</span>

            content <span class="token operator">+=</span> <span class="token template-string"><span class="token template-punctuation string">`</span><span class="token string"> |</span><span class="token interpolation"><span class="token interpolation-punctuation punctuation">${</span>Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token keyword">this</span><span class="token punctuation">.</span>n<span class="token punctuation">]</span><span class="token interpolation-punctuation punctuation">}</span></span><span class="token template-punctuation string">`</span></span>

            <span class="token keyword">return</span> content
        <span class="token punctuation">}</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">join</span><span class="token punctuation">(</span><span class="token string">'\r\n'</span><span class="token punctuation">)</span>

        <span class="token keyword">return</span> result
    <span class="token punctuation">}</span>

    <span class="token function">_forward</span><span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">let</span> n <span class="token operator">=</span> <span class="token keyword">this</span><span class="token punctuation">.</span>m
        <span class="token keyword">let</span> maxRow
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">let</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> n<span class="token punctuation">;</span> i <span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token comment">// Ab[i][i] 为主元</span>
            maxRow <span class="token operator">=</span> <span class="token keyword">this</span><span class="token punctuation">.</span><span class="token function">_maxRow</span><span class="token punctuation">(</span>i<span class="token punctuation">,</span> n<span class="token punctuation">)</span><span class="token punctuation">;</span>
            <span class="token punctuation">[</span><span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">,</span> <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>maxRow<span class="token punctuation">]</span><span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>maxRow<span class="token punctuation">]</span><span class="token punctuation">,</span> <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">]</span><span class="token punctuation">;</span>

            <span class="token comment">// 将主元归一，NOTE:它应该想的是让this.Ab[i] 这一行全都除以这个系数，从而实现主元归一</span>
            <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">.</span><span class="token function">map</span><span class="token punctuation">(</span><span class="token parameter">num</span> <span class="token operator">=&gt;</span> <span class="token function">_d</span><span class="token punctuation">(</span>num<span class="token punctuation">,</span> <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">)</span> <span class="token comment">// this.Ab[i] / this.Ab[i][i] TODO this.AB[i][i] === 0</span>

            <span class="token comment">// 高斯消元</span>
            <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">let</span> j <span class="token operator">=</span> i <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">;</span> j <span class="token operator">&lt;</span> n<span class="token punctuation">;</span> j <span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token keyword">let</span> Abi <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token operator">...</span><span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">]</span>
                Abi <span class="token operator">=</span> Abi<span class="token punctuation">.</span><span class="token function">map</span><span class="token punctuation">(</span><span class="token parameter">num</span> <span class="token operator">=&gt;</span> <span class="token function">_m</span><span class="token punctuation">(</span>num<span class="token punctuation">,</span> <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
                <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">.</span><span class="token function">map</span><span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token parameter">num<span class="token punctuation">,</span> _</span><span class="token punctuation">)</span> <span class="token operator">=&gt;</span> <span class="token function">_c</span><span class="token punctuation">(</span>num<span class="token punctuation">,</span> Abi<span class="token punctuation">[</span>_<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
    <span class="token punctuation">}</span>

    <span class="token function">_backward</span><span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">let</span> n <span class="token operator">=</span> <span class="token keyword">this</span><span class="token punctuation">.</span>m

        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">let</span> i <span class="token operator">=</span> n <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span> i <span class="token operator">&gt;</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span> i <span class="token operator">--</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token comment">// Ab[i][i] 为主元</span>
            <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">let</span> j <span class="token operator">=</span> i <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span> j <span class="token operator">&gt;</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span> j <span class="token operator">--</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token keyword">let</span> Abi <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token operator">...</span><span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">]</span>
                Abi <span class="token operator">=</span> Abi<span class="token punctuation">.</span><span class="token function">map</span><span class="token punctuation">(</span><span class="token parameter">num</span> <span class="token operator">=&gt;</span> <span class="token function">_m</span><span class="token punctuation">(</span>num<span class="token punctuation">,</span> <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
                <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">.</span><span class="token function">map</span><span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token parameter">num<span class="token punctuation">,</span> _</span><span class="token punctuation">)</span> <span class="token operator">=&gt;</span> <span class="token function">_c</span><span class="token punctuation">(</span>num<span class="token punctuation">,</span> Abi<span class="token punctuation">[</span>_<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
    <span class="token punctuation">}</span>

    <span class="token comment">// 找到主元最大的那一行</span>
    <span class="token function">_maxRow</span> <span class="token punctuation">(</span><span class="token parameter">index<span class="token punctuation">,</span> n</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">let</span> <span class="token punctuation">[</span>best<span class="token punctuation">,</span> ret<span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>index<span class="token punctuation">]</span><span class="token punctuation">[</span>index<span class="token punctuation">]</span><span class="token punctuation">,</span> index<span class="token punctuation">]</span><span class="token punctuation">;</span>

        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">let</span> i <span class="token operator">=</span> index <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> n<span class="token punctuation">;</span> i <span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">if</span> <span class="token punctuation">(</span><span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>index<span class="token punctuation">]</span> <span class="token operator">&lt;</span> best<span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token punctuation">[</span>best<span class="token punctuation">,</span> ret<span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token keyword">this</span><span class="token punctuation">.</span>Ab<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>index<span class="token punctuation">]</span><span class="token punctuation">,</span> i<span class="token punctuation">]</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">return</span> ret
    <span class="token punctuation">}</span>

<span class="token punctuation">}</span>


<span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token operator">=&gt;</span> <span class="token punctuation">{</span>
    <span class="token keyword">const</span> <span class="token constant">A</span> <span class="token operator">=</span><span class="token punctuation">[</span>
        <span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">4</span><span class="token punctuation">]</span><span class="token punctuation">,</span>
        <span class="token punctuation">[</span><span class="token number">3</span><span class="token punctuation">,</span> <span class="token number">7</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">]</span><span class="token punctuation">,</span>
        <span class="token punctuation">[</span><span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">3</span><span class="token punctuation">,</span> <span class="token number">3</span><span class="token punctuation">]</span><span class="token punctuation">,</span>
    <span class="token punctuation">]</span>
    <span class="token keyword">const</span> b <span class="token operator">=</span> <span class="token punctuation">[</span>
        <span class="token number">7</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">11</span><span class="token punctuation">,</span> <span class="token number">1</span>
    <span class="token punctuation">]</span>

    <span class="token comment">// const A =[</span>
    <span class="token comment">//     [1, -3, 5],</span>
    <span class="token comment">//     [2, -1, -3],</span>
    <span class="token comment">//     [3, 1, 4],</span>
    <span class="token comment">// ]</span>
    <span class="token comment">// const b = [</span>
    <span class="token comment">//     -9, 19, -13</span>
    <span class="token comment">// ]</span>

    <span class="token comment">// const A =[</span>
    <span class="token comment">//     [1, 2, -2],</span>
    <span class="token comment">//     [2, -3, 1],</span>
    <span class="token comment">//     [3, -1, 3],</span>
    <span class="token comment">// ]</span>
    <span class="token comment">// const b = [</span>
    <span class="token comment">//    6, -10, -16</span>
    <span class="token comment">// ]</span>

    <span class="token comment">// const A =[</span>
    <span class="token comment">//     [8, 2, 1, 2.5],</span>
    <span class="token comment">//     [1, 8,-0.5, 2],</span>
    <span class="token comment">//     [1.5, 2, 8, -1],</span>
    <span class="token comment">//     [1, 0.5, 0.7, 8],</span>
    <span class="token comment">// ]</span>
    <span class="token comment">// const b = [</span>
    <span class="token comment">//     1.5, -3, -4.5, 3.2</span>
    <span class="token comment">// ]</span>

    <span class="token keyword">const</span> linearSystem <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">LinearSystem</span><span class="token punctuation">(</span><span class="token constant">A</span><span class="token punctuation">,</span> b<span class="token punctuation">)</span>
    linearSystem<span class="token punctuation">.</span><span class="token function">gaussJordanElimination</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
    linearSystem<span class="token punctuation">.</span><span class="token function">fancyPrint</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
<span class="token punctuation">}</span><span class="token punctuation">)</span><span class="token punctuation">(</span><span class="token punctuation">)</span>

<span class="token comment">// #region  精确数学运算</span>
<span class="token comment">// 计算的地方很多的话，需要很精确的话，建议使用math.js这个插件库</span>

 <span class="token comment">//加法函数，用来得到精确的加法结果</span>
  <span class="token comment">//说明：javascript的加法结果会有误差，在两个浮点数相加的时候会比较明显。这个函数返回较为精确的加法结果。</span>
  <span class="token comment">//调用：_p(arg1,arg2)</span>
  <span class="token comment">//返回值：arg1加上arg2的精确结果</span>
  <span class="token keyword">function</span> <span class="token function">_p</span><span class="token punctuation">(</span><span class="token parameter">arg1<span class="token punctuation">,</span> arg2</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
    arg1 <span class="token operator">=</span> <span class="token function">parseFloat</span><span class="token punctuation">(</span>arg1<span class="token punctuation">)</span><span class="token punctuation">;</span>
    arg2 <span class="token operator">=</span> <span class="token function">parseFloat</span><span class="token punctuation">(</span>arg2<span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token keyword">var</span> r1<span class="token punctuation">,</span> r2<span class="token punctuation">,</span> m<span class="token punctuation">;</span>
    <span class="token keyword">try</span> <span class="token punctuation">{</span> r1 <span class="token operator">=</span> arg1<span class="token punctuation">.</span><span class="token function">toString</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">split</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">)</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length <span class="token punctuation">}</span> <span class="token keyword">catch</span> <span class="token punctuation">(</span>e<span class="token punctuation">)</span> <span class="token punctuation">{</span> r1 <span class="token operator">=</span> <span class="token number">0</span> <span class="token punctuation">}</span>
    <span class="token keyword">try</span> <span class="token punctuation">{</span> r2 <span class="token operator">=</span> arg2<span class="token punctuation">.</span><span class="token function">toString</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">split</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">)</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length <span class="token punctuation">}</span> <span class="token keyword">catch</span> <span class="token punctuation">(</span>e<span class="token punctuation">)</span> <span class="token punctuation">{</span> r2 <span class="token operator">=</span> <span class="token number">0</span> <span class="token punctuation">}</span>
    m <span class="token operator">=</span> Math<span class="token punctuation">.</span><span class="token function">pow</span><span class="token punctuation">(</span><span class="token number">100</span><span class="token punctuation">,</span> Math<span class="token punctuation">.</span><span class="token function">max</span><span class="token punctuation">(</span>r1<span class="token punctuation">,</span> r2<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token keyword">return</span> <span class="token punctuation">(</span><span class="token function">_m</span><span class="token punctuation">(</span>arg1<span class="token punctuation">,</span> m<span class="token punctuation">)</span> <span class="token operator">+</span> <span class="token function">_m</span><span class="token punctuation">(</span>arg2<span class="token punctuation">,</span> m<span class="token punctuation">)</span><span class="token punctuation">)</span> <span class="token operator">/</span> m<span class="token punctuation">;</span>
  <span class="token punctuation">}</span>
 
  <span class="token comment">//减法函数，用来得到精确的减法结果</span>
  <span class="token comment">//说明：javascript的加法结果会有误差，在两个浮点数相加的时候会比较明显。这个函数返回较为精确的减法结果。</span>
  <span class="token comment">//调用：_c(arg1,arg2)</span>
  <span class="token comment">//返回值：arg1减去arg2的精确结果</span>
  <span class="token keyword">function</span> <span class="token function">_c</span><span class="token punctuation">(</span><span class="token parameter">arg1<span class="token punctuation">,</span> arg2</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
    arg1 <span class="token operator">=</span> <span class="token function">parseFloat</span><span class="token punctuation">(</span>arg1<span class="token punctuation">)</span><span class="token punctuation">;</span>
    arg2 <span class="token operator">=</span> <span class="token function">parseFloat</span><span class="token punctuation">(</span>arg2<span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token keyword">var</span> r1<span class="token punctuation">,</span> r2<span class="token punctuation">,</span> m<span class="token punctuation">,</span> n<span class="token punctuation">;</span>
    <span class="token keyword">try</span> <span class="token punctuation">{</span> r1 <span class="token operator">=</span> arg1<span class="token punctuation">.</span><span class="token function">toString</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">split</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">)</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length <span class="token punctuation">}</span> <span class="token keyword">catch</span> <span class="token punctuation">(</span>e<span class="token punctuation">)</span> <span class="token punctuation">{</span> r1 <span class="token operator">=</span> <span class="token number">0</span> <span class="token punctuation">}</span>
    <span class="token keyword">try</span> <span class="token punctuation">{</span> r2 <span class="token operator">=</span> arg2<span class="token punctuation">.</span><span class="token function">toString</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">split</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">)</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length <span class="token punctuation">}</span> <span class="token keyword">catch</span> <span class="token punctuation">(</span>e<span class="token punctuation">)</span> <span class="token punctuation">{</span> r2 <span class="token operator">=</span> <span class="token number">0</span> <span class="token punctuation">}</span>
    m <span class="token operator">=</span> Math<span class="token punctuation">.</span><span class="token function">pow</span><span class="token punctuation">(</span><span class="token number">10</span><span class="token punctuation">,</span> Math<span class="token punctuation">.</span><span class="token function">max</span><span class="token punctuation">(</span>r1<span class="token punctuation">,</span> r2<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token comment">//动态控制精度长度</span>
    n <span class="token operator">=</span> <span class="token punctuation">(</span>r1 <span class="token operator">&gt;=</span> r2<span class="token punctuation">)</span> <span class="token operator">?</span> r1 <span class="token operator">:</span> r2<span class="token punctuation">;</span>
    <span class="token keyword">return</span> <span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token function">_m</span><span class="token punctuation">(</span>arg1<span class="token punctuation">,</span> m<span class="token punctuation">)</span> <span class="token operator">-</span> <span class="token function">_m</span><span class="token punctuation">(</span>arg2<span class="token punctuation">,</span> m<span class="token punctuation">)</span><span class="token punctuation">)</span> <span class="token operator">/</span> m<span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">toFixed</span><span class="token punctuation">(</span>n<span class="token punctuation">)</span><span class="token punctuation">;</span>
  <span class="token punctuation">}</span>
 
  <span class="token comment">//乘法函数，用来得到精确的乘法结果</span>
  <span class="token comment">//说明：javascript的乘法结果会有误差，在两个浮点数相乘的时候会比较明显。这个函数返回较为精确的乘法结果。</span>
  <span class="token comment">//调用：_m(arg1,arg2)</span>
  <span class="token comment">//返回值：arg1乘以arg2的精确结果</span>
  <span class="token keyword">function</span> <span class="token function">_m</span><span class="token punctuation">(</span><span class="token parameter">arg1<span class="token punctuation">,</span> arg2</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
    arg1 <span class="token operator">=</span> <span class="token function">parseFloat</span><span class="token punctuation">(</span>arg1<span class="token punctuation">)</span><span class="token punctuation">;</span>
    arg2 <span class="token operator">=</span> <span class="token function">parseFloat</span><span class="token punctuation">(</span>arg2<span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token keyword">var</span> m <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">,</span> s1 <span class="token operator">=</span> arg1<span class="token punctuation">.</span><span class="token function">toString</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">,</span> s2 <span class="token operator">=</span> arg2<span class="token punctuation">.</span><span class="token function">toString</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token keyword">try</span> <span class="token punctuation">{</span> m <span class="token operator">+=</span> s1<span class="token punctuation">.</span><span class="token function">split</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">)</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length <span class="token punctuation">}</span> <span class="token keyword">catch</span> <span class="token punctuation">(</span>e<span class="token punctuation">)</span> <span class="token punctuation">{</span> <span class="token punctuation">}</span>
    <span class="token keyword">try</span> <span class="token punctuation">{</span> m <span class="token operator">+=</span> s2<span class="token punctuation">.</span><span class="token function">split</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">)</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length <span class="token punctuation">}</span> <span class="token keyword">catch</span> <span class="token punctuation">(</span>e<span class="token punctuation">)</span> <span class="token punctuation">{</span> <span class="token punctuation">}</span>
    <span class="token keyword">return</span> <span class="token function">Number</span><span class="token punctuation">(</span>s1<span class="token punctuation">.</span><span class="token function">replace</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">,</span> <span class="token string">&quot;&quot;</span><span class="token punctuation">)</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token function">Number</span><span class="token punctuation">(</span>s2<span class="token punctuation">.</span><span class="token function">replace</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">,</span> <span class="token string">&quot;&quot;</span><span class="token punctuation">)</span><span class="token punctuation">)</span> <span class="token operator">/</span> Math<span class="token punctuation">.</span><span class="token function">pow</span><span class="token punctuation">(</span><span class="token number">10</span><span class="token punctuation">,</span> m<span class="token punctuation">)</span><span class="token punctuation">;</span>
  <span class="token punctuation">}</span>
 
  <span class="token comment">//除法函数，用来得到精确的除法结果</span>
  <span class="token comment">//说明：javascript的除法结果会有误差，在两个浮点数相除的时候会比较明显。这个函数返回较为精确的除法结果。</span>
  <span class="token comment">//调用：_d(arg1,arg2)</span>
  <span class="token comment">//返回值：arg1除以arg2的精确结果</span>
  <span class="token keyword">function</span> <span class="token function">_d</span><span class="token punctuation">(</span><span class="token parameter">arg1<span class="token punctuation">,</span> arg2</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
    arg1 <span class="token operator">=</span> <span class="token function">parseFloat</span><span class="token punctuation">(</span>arg1<span class="token punctuation">)</span><span class="token punctuation">;</span>
    arg2 <span class="token operator">=</span> <span class="token function">parseFloat</span><span class="token punctuation">(</span>arg2<span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token keyword">var</span> t1 <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">,</span> t2 <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">,</span> r1<span class="token punctuation">,</span> r2<span class="token punctuation">;</span>
    <span class="token keyword">try</span> <span class="token punctuation">{</span> t1 <span class="token operator">=</span> arg1<span class="token punctuation">.</span><span class="token function">toString</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">split</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">)</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length<span class="token punctuation">;</span> <span class="token punctuation">}</span> <span class="token keyword">catch</span> <span class="token punctuation">(</span>e<span class="token punctuation">)</span> <span class="token punctuation">{</span> <span class="token punctuation">}</span>
    <span class="token keyword">try</span> <span class="token punctuation">{</span> t2 <span class="token operator">=</span> arg2<span class="token punctuation">.</span><span class="token function">toString</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">split</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">)</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length<span class="token punctuation">;</span> <span class="token punctuation">}</span> <span class="token keyword">catch</span> <span class="token punctuation">(</span>e<span class="token punctuation">)</span> <span class="token punctuation">{</span> <span class="token punctuation">}</span>
    r1 <span class="token operator">=</span> <span class="token function">Number</span><span class="token punctuation">(</span>arg1<span class="token punctuation">.</span><span class="token function">toString</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">replace</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">,</span> <span class="token string">&quot;&quot;</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
    r2 <span class="token operator">=</span> <span class="token function">Number</span><span class="token punctuation">(</span>arg2<span class="token punctuation">.</span><span class="token function">toString</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">.</span><span class="token function">replace</span><span class="token punctuation">(</span><span class="token string">&quot;.&quot;</span><span class="token punctuation">,</span> <span class="token string">&quot;&quot;</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token keyword">return</span> <span class="token function">_m</span><span class="token punctuation">(</span>r1 <span class="token operator">/</span> r2 <span class="token punctuation">,</span> Math<span class="token punctuation">.</span><span class="token function">pow</span><span class="token punctuation">(</span><span class="token number">10</span><span class="token punctuation">,</span> t2 <span class="token operator">-</span> t1<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
  <span class="token punctuation">}</span>

<span class="token comment">//  #endregion</span>
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